Induced representation of a locally compact group

Summary

Unitary induction builds a unitary representation of a locally compact group $G$ from a unitary representation of a closed subgroup $H$. The new space consists of square-integrable vector-valued functions on $G$ that transform under $H$ by the given representation, and $G$ acts by translation. It extends the induced representation of finite groups to subgroups of infinite index, such as lattices and parabolic subgroups. Mackey's theory of induced representations uses it to build and classify irreducible representations, for example the principal series of semisimple Lie groups.